Interpolation and Sampling: E.T. Whittaker, K. Ogura and Their Followers

Butzer, Paul L.; Ferreira, Paulo J. S. G.; Higgins, John R.; Saitoh, Saburou; Schmeißer, Gerhard; Stens, R. L.

doi:10.1007/s00041-010-9131-8

Cited by 51 publications

(13 citation statements)

References 40 publications

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“…Here {d n } may, for example, arise from normalizing factors, as e.g. in (5). Let T := {t n } ⊂ E, (n ∈ X), denote a set of distinct points of E. When it is supposed that T is such that {κ tn } is an orthogonal set in H we can hardly expect it to come already normalized, since κ t must be defined before the assumption concerning T is made.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

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Converse Sampling and Interpolation

Higgins¹

2015

STSIP

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Section: Preliminaries and Notationsmentioning

confidence: 99%

“…Actually, Ogura presented it as being converse to a theorem of E.T. Whittaker but there is some obscurity surrounding this (see [5]).…”

Section: Introductionmentioning

confidence: 99%

Converse Sampling and Interpolation

Higgins¹

2015

STSIP

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“…The sinc approximation is "bandlimited" in the sense that F T (f localized−sinc (k; h) is zero for all |k| > π/h. The Shannon-Kotelnikov-Whittaker Sampling Theorem [12] states that if f(x) is bandlimited, that is, if F (k) = 0 for all |k| ≥ π/h, the sinc approximation with grid spacing h is exact. These facts, long known in signal processing, imply that all the error for a non-bandlimited function must come entirely from F (k) for |k| > π/h.…”

Section: Sinc Pseudospectral Errorsmentioning

confidence: 99%

A Fourier error analysis for radial basis functions and the Discrete Singular Convolution on an infinite uniform grid, Part 1: Error theorem and diffusion in Fourier space

Boyd

2015

Applied Mathematics and Computation

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On an infinite grid with uniform spacing h, the cardinal basis Cj (x; h) for many spectral methods consists of translates of a "master cardinal function", Cj (x; h) = C(x/h − j). The cardinal basis satisfies the usual Lagrange cardinal condition, Cj(mh) = δjm where δjm is the Kronecker delta function. All such "shift-invariant subspace" master cardinal functions are of "localized-sinc" form in the sense that C(X) = sinc(X)s(X) for a localizer function s which is smooth and analytic on the entire real axis and the Whittaker cardinal function is sinc(X) ≡ sin(πX)/(πX). The localized-sinc approximation to a general f (x) is f localized−sinc (x; h) ≡ P ∞ j=−∞ f (jh) s([x − jh]/h) sinc([x − jh]/h). In contrast to most radial basis function applications, matrix factorization is unnecessary. We prove a general theorem for the Fourier transform of the interpolation error for localized-sinc bases. For exponentially-convergent radial basis functions (RBFs) [Gaussians, inverse multiquadrics, etc.] and the basis functions of the Discrete Singular Convolution (DSC), the localizer function is known exactly or approximately. This allows us to perform additional error analysis for these bases. We show that the error is similar to that for sinc bases except that the localizer acts like a diffusion in Fourier space, smoothing the sinc error.

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“… and is thus often known as the Shannon sampling theorem. However, there is evidence showing that other authors had been previously involved in this subject such as, for instance, Kotelnikov , Whittaker , and others [8,16]. Shannon's work, however, had a great impact on the adoption of the sampling theorem by the engineering community as a practical engineering tool.…”

Section: Introductionmentioning

confidence: 99%

An interactive approach for illustrating a proof of the sampling theorem using MATHEMATICA

Guerrero¹

2018

Comp Applic In Engineering

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Learning methods in engineering education should evolve to take advantage of the progress of technology. Despite being one of the most important theorems of engineering, at undergraduate level the foundations of sampling theorem are usually oversimplified, whereas at graduate level its proof is frequently presented in the context of a rigorous mathematical framework. This paper presents an interactive approach to illustrate an amenable proof of the sampling theorem through example. Apart from its practical value as a learning tool, the proof captures some mathematical subtleties that can help to understand some more advanced concepts. The goal of this work is to show that learning the sampling theorem through an interactive example of its mathematical proof can be an enjoyable and insightful experience.

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Interpolation and Sampling: E.T. Whittaker, K. Ogura and Their Followers

Cited by 51 publications

References 40 publications

Converse Sampling and Interpolation

Converse Sampling and Interpolation

A Fourier error analysis for radial basis functions and the Discrete Singular Convolution on an infinite uniform grid, Part 1: Error theorem and diffusion in Fourier space

An interactive approach for illustrating a proof of the sampling theorem using MATHEMATICA

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